You know, back before APIs were all the rage, people wrote clients for web services by scraping. Twitter really wouldn't be able to do anything about a Twitter client that pretended to be IE9.
I'm sure Twitter knows how to block crawlers, especially ones that they're actually aware of.
Since the client runs on the user's machine, performance isn't really an issue, either. (Trying to do it for all users on a server is a different story.)
APIs are indeed "all the rage" because they're easier and don't change as much. But when given no other option... You do what you have to.
Not as straightforward as an API to be sure, but not impossible by a long shot.
A business? Probably too high risk. A nice FOSS client, though? Why not?